Let's replace the word "space" with "manifold" because its more general.
A Riemannian manifold is a manifold having a positive definite metric.
A Euclidean manifold is a special case of a Riemannian manifold where the positive definite metric is a Euclidean metric i.e. [itex] d(x,y)=\sqrt{\sum_i (x_i-y_i)^2} [/itex].

About your second question,the semi-definite metric making our manifold a Riemannian one,maybe not induced by an inner product!
Also the metric space in question maybe not complete.
So no,not all Riemannian manifolds are Hilbert Spaces!
But it seems to me that every Real Hilbert Space,is a Riemmanian manifold!
(Sorry math people for putting my feet into your shoes!)

Let's replace the word "space" with "manifold" because its more general.

Unless by space you mean something like vector spaces, it's actually the other way around. The restriction to manifolds is necessary, however, since they are precisely the spaces on which Riemannian metrics are defined.

A Riemannian manifold is a manifold having a positive definite metric.
A Euclidean manifold is a special case of a Riemannian manifold where the positive definite metric is a Euclidean metric

They are technically different kinds of metrics. The metrics you learn about when studying metric spaces are very different than Riemannian metrics.

Unless by space you mean something like vector spaces, it's actually the other way around. The restriction to manifolds is necessary, however, since they are precisely the spaces on which Riemannian metrics are defined.
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In fact I was considering the "space" in the OP to mean 3-dimensional Euclidean manifold!

They are technically different kinds of metrics. The metrics you learn about when studying metric spaces are very different than Riemannian metrics.

I was starting to feel that way too,because the wikipedia page on Riemannian manifolds were defining Riemannian metrics somehow that I couldn't relate it to the definition of metric in metric spaces!
So I retreat and leave this thread to mathematicians.

What's the difference between Euclidean and Riemann space? As far as I know ##\mathbb{R}^n## is Euclidean space.

Riemannian manifolds are those manifolds equipped with a specific Riemannian metric. It can be shown that every manifold can be endowed with such a metric.

Euclidean space has a bit more flexible interpretation in my opinion. Sometimes it can refer to Rn purely as a topological space. Other times it may refer to the vector space structure of Rn. It could mean a combination of the two as well. Or it could refer to Rn as a Riemmanian manifold with the usual metric or something else still.